Most high-school level textbooks that I am familiar with do refer to the War of 1812 as the 2nd American Revolution. The reasons stated in previous posts are excellent and I will not repeat them here.
However, I would like to give, perhaps, a diffrerent perspective I believe to be true but is simply not generally mentioned in texts. Consider this; A genuine revolution may very well have been fought by the Northern troops during the Civil War, at least after January 1, 1863. After that date (the effective date of the Emancipation Proclamation), the abolishment of slavery became a Northern goal. Since slavery had long been established and sanctioned by the Constitution (references to the slave trade, the 3/5 Compromise, etc), and supported by the US Supreme Court (Dred Scott), an attempt by the government to abolish that institution was unconstitutional and therefore, decidedly revolutionary.
All the South was doing was fighting a rebellion.
Two American Revolutions:It is common in the scholarly literature on the founding of the American republic, and in college courses based on that literature, to refer to the Philadelphia Convention of 1787 and the adoption in 1789 of the constitution produced by that convention, as the second American Revolution. See Edward Countryman's chapter 6 "One Republic" in his book The American Revolution (New York: Hill and Wang, 1985) for a good summary of this view.
This second revolution countered many aspects of the first revolution. Such as, the popular democracy that was written into many of the state constitutions and dominated the nation through the Articles of Confederation, was countered with representative democracy. For example, under the new constitution the voters elected an electoral college which elected the U. S. President; the voters elected a state legislature which then elected the U.S. Senators; the U.S. Representatives were electied from such large districts that it was thought only big men who were very well known could be elected; state constitutions provided for direct election by most free men of most officers. Such as implementing one national commercial policy as replacement for 13 separate state commercial policies. Hamilton and the new capitalists of America saw that they would not be able to take fullest advantage of all of the machinations and dissimilations and duplicities for transferring wealth and creating imaginary wealth on paper, that are characteristic of capitalism, unless they controlled the nations economic policies from a national government. Also, the nation's tremendous potential for creating wealth through trade could not be realized under 13 different commercial policies, so the government under the new Constitution was given power to regulate interstate commerce.
Some of the other answers are confusing "revolution" and "war of independence." The War of 1812 is sometimes referred to as our second war of independence. A revolution is a change in government. Our first one was effected by the Declaration of Independence and the establishment of the government under the Articles of Confederation to replace the King and Parliament. Our second was effected by the framing and adoption of the Constitution of 1789, which changed our government rather much.
A third revolution was that one promolgated by the Republican Party and the Industrial Social System in the 1850s-'70s. This is the revolution that the Southern States revolted against. Theirs was a counter-revolution against the revolution was being promoted by the North.
Finally, a rebellion is not a revolution until it is successful in changing the government. Furthermore, a rebellion is opposition to lawful authority. It was the victory of the Radical Republican Party in the American War of 1861-1865, that made the states over into provinces of the national government. Prior to that time, the national government was the creature of the states, and it was not illegal for states to seceed; if a state had remained within the union and defied the lawful and constitutional powers of the national government, that would have been rebellion, but a state's withdrawal from a government that it had been a party to creating and which was created as that state's servant, not the state created as that government's servant, was not rebellion. It was the Republican victory that made it possible to say the Southern states rebelled. Had the Confederacy won the war, we would still today be saying that the act of secession was legal and not rebellion.
How were the two American revolutions related? On this, I am less sure than on the first part. The state governments under the Articles of Confederation were more powerful than the Confederation government and were controlled by more liberal and more middle-class people. The conservatives and large merchants and large land-owners, thought that society would be more stable and commerce more secure if the central government was strong enough to serve as a check on those who controlled the state governments (and naturally they expected themselves to be in charge of the central government). So they set about to change the government, finally accomplishing this with the Philadelphia Convention and the adoption of the Constitution. The leaders of the second revolution had also been among the leaders of the first revolution. Regulation of commerce was a big issue in both revolutions. The first one removed it from the block of politicians in Parliament, representing the English merchants. The second one placed it more firmly in the control of the large American merchants.
If we are taking the American Revolution and the War of 1812 as the two revolutions in question, we would have to make the most natural connection that the presence of the British was something for which military conflict was the only solution. The removal of the influence of the British in the Revolutionary War and their alliance with the Native Americans in the War of 1812 was one of the strongest connections between both conflicts. The presence of economic control and economic rights was also another motivating factor in both conflicts. In both of them, economic rights and the ability to keep and make money was extremely important to both conflicts.
It can refer to either the War of 1812 or the Civil War, as the above posts suggest, depending on who is calling it a revolution.
Many historians will call the War of 1812 America's War for Economic Independence, as the principle reason it was fought was for freedom of the seas/freedom of trade. It was also after this stalemated war that Britain made peace with our independence.
If I was a southerner back then, or perhaps even now, I might call the Civil War the Second American Revolution.
In the first case, the War of 1812 is related to the Revolution because it was largely about issues that were unresolved from the 1783 Treaty of Paris.
In the case of the Civil War, the failure of the Constitution to successfully address the issue of slavery long term meant that sooner or later, a Civil War would have to be fought. It made the war an inevitability. Southerners believed they were merely assuring the rights that had originally been won in the Revolution.
The most usual thing to cite as a second American Revolution is the War of 1812. The reasoning there is that the later war was the one that really consolidated the US's independence.
After the Revolutionary War, the British did not treat the US like a truly independent nation. They kept forts on US territory and they "impressed" American sailors into their navy forcibly. The War of 1812 was fought over these issues.
Because the US did not lose that war, the British started to see them as a truly separate country. They removed their forts and stopped impressing American sailors. This is why that was a second revolution -- it made the US independent in fact as well as in law.
I don't know exactly, but I think it is possible to categorize the revolution against the British as the first and the Civil War as the second, given that the southern states rebelled against the federal government, technically fitting the definition of a revolution.
Both were driven in many ways by economic forces and powerful people who wanted to make sure that their way of making a profit was protected. There were of course other factors, the popular idea of freedom from tyranny, etc., that was common in the run-up to the Revolutionary War and then the ideas of slavery and other things that became embroiled in the Civil War and are often cited as secondary causes.
The term revolution has two related, yet distinct meanings. A general meaning of revolution is some basic and and widespread change taking in societies taking place rapidly. In this sens revolution can be contrasted with "evolution". These revolutions may occur in many different areas such as national government, economics, culture, and technology. Rapid development of technology and and industry in eighteenth and nineteenth century is often referred as "industrial revolution". Also, it is said that an information revolution is currently happening around the whole world.
In a more restricted way the term revolution is used to refer to fundamental change in the character of a nation's government that occurs over a short period because of mass movement rather than because of wars between contending rulers. Such a change may or may not be violent. Most common examples of this type of revolution are American, Russian, and French revolutions.
As per the commonly understood terminology, America experienced only one revolution, the American Revolution, which led to the formation of a new nation called United States of America. There is no other event which is commonly recognized as a second revolution in America. Different authors, experts or even laymen may describe different evens as Second American Revolution because of the importance they attach to such events. The previous two posts illustrate this tendency of individuals to label events as Revolution based on their individual perception.